Electromagnetic Induction

Step 1: Understand resonance in LCR circuits. When an inductor and capacitor are in resonance, their reactances cancel each other out, and the impedance of the circuit becomes purely real, meaning there is no imaginary component to the impedance.
Step 2: Conclusion. Thus, the resonance condition occurs when the inductance and capacitance reactances cancel out, making the impedance purely real.

Step 1: Definition of RMS value. The RMS (Root Mean Square) value of an alternating current (AC) is the effective value of the current. For a sinusoidal wave, it is given by:
Step 2: Conclusion. Thus, the RMS value is .

Step 1: Understanding resonance in LCR circuits. The resonance in an LCR circuit occurs when the reactances of the inductor and capacitor cancel each other out, and the impedance of the circuit is at a minimum. This results in the sharpest resonance when the resistance is small.
Step 2: Conclusion. Thus, the resonance is sharpest when is small.

Step 1: Lenz’s Law and Faraday’s Law of Induction. The induced EMF is maximum when the change in magnetic flux is greatest. This happens when the plane of the coil is perpendicular to the magnetic field.
Step 2: Conclusion. Thus, the induced EMF is maximum when the coil's plane is at 45° to the magnetic field.

Step 1: Frequency dependence. The frequency of an AC supply determines the inductive and capacitive reactances in the circuit. These reactances depend on the frequency and influence the behavior of the circuit.
Step 2: Conclusion. Hence, the frequency is determined by both inductive and capacitive reactance.

Step 1: Resonance condition.
At resonance in an LCR circuit, the inductive reactance and capacitive reactance cancel out. The resonance frequency is given by: Substituting the values of and , we get the resonance frequency.

Step 2: Conclusion.
Thus, the resonance frequency is Hz. Hence, the correct answer is option (B).

Final Answer:

Step 1: Formula for induced emf.
The emf induced in the secondary coil is given by: where is the mutual inductance, is the change in current, and is the time interval. Substituting the given values, we get the induced emf.

Step 2: Conclusion.
Thus, the induced emf in the secondary coil is V. Hence, the correct answer is option (B).

Final Answer:

Step 1: Condition for maximum power.
Maximum power in an LCR circuit is dissipated at resonance frequency, where . The resonance frequency is given by: Substituting the given values, we calculate the frequency.

Step 2: Conclusion.
Thus, the resonance frequency is approximately 50.3 Hz. Hence, the correct answer is option (D).

Final Answer:

Step 1: Understanding Faraday's Law.
The induced current is maximum when the magnetic flux through the coil is changing at the fastest rate. This happens when the plane of the coil is parallel to the magnetic field, which results in the maximum flux.

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer:

Step 1: Analyzing the total inductance.
When solenoids are connected in series, the total inductance depends on the configuration. When they are placed with the same or opposite senses of the turns, it results in different total inductance. In case 1, the inductance is simply the sum of the individual inductances, while in case 2 and case 3, the inductance depends on how they are arranged (same or opposite direction).

Step 2: Conclusion.
Thus, the correct inductance values for each case are 2L, 4L, and 4L respectively. Therefore, the correct answer is option (D).

Final Answer:

Step 1: Recall factors affecting self-inductance.
Self-inductance depends on geometry of coil:
Number of turns, cross-sectional area, length, and permeability of medium.

Step 2: Compare coil 1 and coil 2.
Coil 2 has reversed winding direction in each layer, but total turns and geometry remain same.
Self-inductance depends on total flux linkage due to current, not on resistance or winding direction of different layers.
So:

Step 3: Effect of superconducting wire on inductance.
Superconducting wire changes resistance (becomes zero), but inductance depends on geometry and magnetic flux linkage.
So:

Step 4: Final conclusion.
All coils have same inductance.
Final Answer:

Step 1: Understand electromagnetic induction in moving conductor.
When a metallic plate moves between magnetic poles, magnetic flux linked with the plate changes continuously.
Step 2: Eddy currents are induced.
Due to changing flux, circulating currents are induced within the plate. These are called eddy currents.
Step 3: Apply Lenz's Law.
By Lenz’s law, the induced currents produce a magnetic field that opposes the change producing them.
So the plate experiences a force opposite to its motion (magnetic damping).
Step 4: Conclusion.
Hence eddy currents are produced and they oppose the motion of the plate.
Final Answer:

Step 1: Clarify what happens when switch is turned ON.
Electrons in wires already exist, but when switch is closed, an electric field is established throughout the circuit.
Step 2: Speed of electrons vs speed of signal.
The drift velocity of electrons is very slow (order of to ).
So electrons do not move at speed of light.
Step 3: Signal propagation.
The information that current should start flowing is carried by electromagnetic waves (electric field + magnetic field) travelling through conductor at nearly speed of light.
Step 4: Conclusion.
Hence appliance responds immediately because signal travels very fast.
Final Answer:

Step 1: Use transformer turns-voltage relation.
For an ideal transformer:

Step 2: Substitute given voltages.
Primary voltage .
Secondary voltage .
Step 3: Calculate turns ratio.

Step 4: Match with option.
Nearest value is .
Final Answer:

Step 1: Read the given values.
From circuit diagram:



Given:

Step 2: Compute inductive reactance.


Step 3: Compute capacitive reactance.


Step 4: Compare and .
Since:

Net reactance:

So circuit behaves capacitive.
Thus it behaves like a series R-C circuit.
Final Answer:

Step 1: Use Maxwell’s displacement current concept.
Between capacitor plates, magnetic field is due to displacement current.
Using Ampere-Maxwell law:
Step 2: Electric flux through area.
Step 3: Substitute in equation.
Step 4: Insert values.
, .
Final Answer:

Step 1: Use Faraday’s law in terms of charge.
Induced emf:
Current:
Charge flowed:
Step 2: Change in flux when rotated by .
Initial flux:
Final flux after :
So,
Magnitude:
Step 3: Substitute in charge formula.
Final Answer:

Step 1: Use Faraday’s Law to calculate emf.
The induced emf is given by: where is the number of turns, and is the magnetic flux.
Step 2: Explanation.
After applying the formula, we find the induced emf in the coil to be .
Final Answer:

Step 1: Use the formula for induced emf.
The induced emf in the coil is given by Faraday's law of induction: where is the magnetic flux.
Step 2: Calculate the charge.
The amount of charge is given by: Substituting the values and solving for , we get .
Final Answer:

Step 1: Use the formula for energy stored in an inductor.
The energy stored in an inductor is given by: where is the inductance and is the current.
Step 2: Calculate the current.
The current is given by: Substituting the values into the energy formula, we get:
Final Answer:

The induced emf is related to the rate of change of current in the coil. The time variation of the current leads to a time-varying magnetic field that induces an emf in the coil, which follows the time variation of the current.

Step 2: Conclusion.
The induced emf varies with time in the same manner as the current, corresponding to option (d).

In a transformer, the current is related by the ratio , where and are the number of turns in the primary and secondary coils, respectively. Using this relation, we find the current through the load resistance.

Step 2: Conclusion.
The current through the load resistance is 2A, corresponding to option (c).

The induced emf in the second coil is given by Faraday's law of induction. The maximum value of emf can be found using the formula: where is the mutual inductance and is the rate of change of current.

To prevent sparking, the capacitor needs to absorb the energy stored in the inductor when the circuit is switched off. The capacitance needed can be calculated using the energy balance between the inductor and the capacitor.

The self-inductance of a solenoid is given by the formula: Where is the permeability of free space, is the relative permeability, is the number of turns, is the cross-sectional area, and is the length of the solenoid.

The energy stored in an inductor is given by the formula: Where is the inductance and is the current. First, calculate the current using Ohm's law, and then use it to find the energy stored in the inductor.

According to Lenz's law, the induced emf always opposes the change in the magnetic field that caused it. This means the induced emf has a direction opposite to the applied field direction.

Using the formula for magnetic flux , where , , and , we calculate the flux as .

Step 1: Time Constant in Circuit.
The time constant of an circuit is given by: The current reaches of its steady state value in 4 seconds, which is approximately the time constant . Thus, the time constant is seconds.
Step 2: Conclusion.
The correct answer is (B), sec.

Step 1: Faraday's Law.
According to Faraday's law of induction, the induced emf is given by: where is the magnetic flux. Substituting the given flux equation , we get: Step 2: Determining the Direction Change.
For the current to change direction, the sign of the induced emf must change. Solving for , we get the time at which this change occurs at . Step 3: Conclusion.
The correct answer is (B), 0.5 sec.

Step 1: Understanding Induction.
According to Faraday's Law of Induction, a changing magnetic flux through a closed loop induces an electromotive force (emf) in the loop. As the loop moves towards the current-carrying wire, the magnetic flux changes, inducing a current.
Step 2: Determining the Direction of the Induced Current.
Using the right-hand rule, we determine that the induced current will be clockwise to oppose the change in magnetic flux.
Step 3: Conclusion.
The correct answer is (B), current is induced in the loop clockwise.

Step 1: Formula for Mutual Inductance.
The formula for emf induced due to a changing current in an inductor is: where is the mutual inductance, is the change in current, and is the time interval.
Step 2: Substituting the Given Values.
Substituting the given values: Solving for , we get . Step 3: Conclusion.
The correct answer is (B), 10mH.

Step 1: Use the formula for induced EMF.
The induced e.m.f. in a coil is given by Faraday’s law of induction: where , and the current changes from 1 A to 11 A in .
Step 2: Calculate the induced e.m.f.
The rate of change of current is: Thus, the induced e.m.f. is:
Final Answer:

Step 1: The maximum e.m.f. induced in the second coil is given by the formula , where is the mutual inductance and is the rate of change of current.
Step 2: The current is given by , so . The maximum value occurs when .
Step 3: The maximum e.m.f. is:
Final Answer:

Step 1: The time constant of an RL circuit is given by the formula , where is the inductance and is the resistance.
Step 2: Given and , the time constant is:
Final Answer:

Induced emf produced between the centre and a point on the disc is given by Putting the values, and We get

Step 1: The induced emf in a rotating disc is given by the formula: where is the magnetic field, is the angular velocity, and is the radius of the disc.
Step 2: Substituting the given values , , and , we get:
Final Answer:

When a bar magnet falls through a conducting ring, an induced current is produced in the ring due to Lenz's law, which opposes the motion of the magnet. This reduces the acceleration of the magnet compared to free fall, making the acceleration less than .
Final Answer:

In an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and to the direction of propagation. This means the energy flows in a direction perpendicular to both the electric and magnetic fields.
Final Answer: